Is there a relationship between the multiplicity of an index and the "algebraic" multiplicity of a zero of a section from a (complex) vector bundle? I'm a physicist who is trying to make sense of the relationship between the number of zeros of a section from an associated vector bundle and the Euler characteristic. My interest lies in applications to gauge theories in which finite energy solutions can be classified according to a topological charge (which is a Chern number/topological degree in the cases I know, and can be related to the Euler characteristic). More precisely, let us consider theorem 11.17 in the book Differential forms ins algebraic Topology from Bott-Tu:

Let $\pi:E\to M$ be an oriented rank k vector bundle over a
compact oriented manifold of dimension k. Let s be a section of E with a finite
number of zeros. The Euler class of E is Poincare dual to the zeros of s,
counted with the appropriate multiplicities.

I have sometimes found it stated that the topological degree counts the number of zeros of a section, with multiplicity. I believe the above theorem is the mathematical justification for this (do correct me if I am wrong, please). But the multiplicity this theorem is talking about is "the local degree of x as a singularity of the section $s/||s||$ of the unit sphere bundle of E relative to some Riemannian structure on E", according to the authors. I wanna know when (if ever) there exists a relationship between this meaning of multiplicity and that found in, say, complex analysis (that's what I called algebraic multiplicity in the title). I will assume the zeros are isolated.
My question was motivated by the behavior of axially symmetric magnetic vortices in the static case of the Nielsen-Olesen/Ginzburg-Landau theory. Here we have a scalar field $\varphi:\mathbb{R^2}\to\mathbb{C}$, seen as a section of the line bundle. Axially symmetric solutions can be taken in the form $\varphi=f(r)e^{in\theta}$, where $n$ is the (integer) topological degree. This field is coupled to a connection $A_{\theta}=A_{\theta}(r)$. The boundary conditions ensure those fields are nonsingular and that the magnetic flux is proportional to $n$. $\varphi$ must have a zero at the origin (and nowhere else). The multiplicity of this zero, in the sense described by Bott-Tu, is indeed $n$, and it may be verified that $f(r)\propto r^n$ to leading order in its power series expansion, so $f(r)$ has a zero of multiplicity $n$. This sounds a lot like the Argument Principle, with the difference that $\varphi$ does not have a complex domain. If $r$ and $\theta$ could be seen as polar coordinates in the complex plane, then this would be a zero of multiplicity $n$ in $\mathbb{C}$. The exact same $r^n$ behavior appears in all vortex theories I know, like [Maxwell or pure] Chern-Simons and many other generalized models, some very different from Nielsen-Olesen.
I'm interested in developing models in gauge theories such as the aforementioned ones (with the exact same topology, bundle and covariant derivative, but different equations of motion), and would like to know if I should expect such a behavior to occur in the solutions to such theories.
Could I find a solution where, for example, $f(r)\propto r^m$, where $m\neq n$ or would that somehow lead to a problem in my theory? Can the degree be used to predict anything about the multiplicity of the zeros of $f(r)$?
Edit (because I accidentally pressed enter before finishing the current bounty description, and didn't find a way to edit that description): While a very good answer has been provided, I still haven't been able to figure out (even after reading some of the references) if something like $f(r)\propto r^m$ could be obtained as a leading order approximation near the origin, with $m\neq n$. The answer to that might be implicit from the current answer, but I can't see it. A definite answer to the last paragraph preceding this edit is sufficient for the reward (although any information concerning sections with a form more general than the proposed $f(r) e^{in\theta}$ will be greatly appreciated as well).
 A: Comment: "I'm a physicist who is trying to make sense of the relationship between the number of zeros of a section from an associated vector bundle and the Euler characteristic."
Reply: There is a theorem (Theorem 5.3) In Harris/Eisenbud's book on algebraic geometry ("3264 and all that...") which relate the chern class $c_i(E)$ of a finite rank locally free sheaf $E$ to the "degeneracy loci" $[D(\tau_i)]$ of a set  of global section of $E$:
Theorem: If $E$ is a locally free sheaf on $X$ of rank $r$ and if $\tau_0,..,\tau_{r-i}\in H^0(X,E)$ are global sections with $D(\tau_i)$ the degeneracy locus where they are independent, then there is an equality
$$c_i(E)=[D(\tau_i)]\in \operatorname{CH}^i(X).$$
Here $\operatorname{CH}^i(X)$ is the i'th Chow group of $X$.
The theorem is valid in general and should apply to an "algebraic variety" over the real or complex numbers. Hence if your manifold is algebraic, this result gives a relation between characteristic classes and degeneracy loci. I believe characteristic classes in various cohomology theories were initially defined using this type of method - it is a more "geometric approach" to this construction.
You may for any non-singular variety $X$ of dimension $n$ define the Euler characteristic of $X$ as follows:
$$ \chi(X):= c(T_X) \cap [C] = \int_X c_n(T_X)$$
where $T_X$ is the tangent bundle. Choose a global section $\tau_0$ of $T_X$ (you view a global section of $T_X$ as a global algebraic vector field on $X$). You get an equality
$$c_n(T_X):=[D(\tau_0)]\in \operatorname{CH}^n(X)$$
and this relates the Euler characteristic $\chi(X)$ to the degeneracy loci $D(\tau_0)$ of a vector field $\tau_0$.
There is a "cycle map"
$$ \gamma: \operatorname{CH}^*(X) \rightarrow H^*(X, \mathbb{Z})$$
where $H^*$ denotes singular cohomology, and you get an equality
$$ c_i^H(E):=\gamma(c_i(E))=\gamma([D(\tau_i)])\in H^i(X, \mathbb{Z}).$$
Hence you get a similar equality in $H^*$.
Remark: In the book of Eisenbud/Harris they speak about Weil divisors/Cartier divisors and invertible sheaves and also the projective bundle formula: For any rank  $e+1$ locally trivial sheaf $E$ on a scheme $X$, they prove there is an isomorphism of rings
$$ \operatorname{CH}^*(\mathbb{P}(E^*)) \cong \operatorname{CH}^*(X)[t]/(t^{e+1})$$
They also speak of the relation between divisors and global sections of invertible sheaves and this exposition is a bit "more detailed" than the one in Hartshorne. The projective bundle is a surjective morphism
$$\pi: \mathbb{P}(E^*) \rightarrow X$$
with the property that for any point $x\in X$ it follows the fiber
$$\pi^{-1}(x)\cong \mathbb{P}^e_{\kappa(x)}$$
is projective $e$-space on the residue field $\kappa(x)$. If $X$ is a complex projective manifold and $x$ is a closed point it follows the fiber is complex projective $e$-space. Hence $\pi$ is a fibration with fibers projective spaces of the same dimension. Some authors use the projective bundle formula to define Chern classes in the Chow ring. Hence the Theorem referred to above says that these definitions give the same result. The "degeneralcy loci approach" is more geometric, the "projective bundle formula approach" is more abstract - if you need this result in your research you should of course convince yourself that the result is correct. It is a classical and much used result.
Example. If $C:=\mathbb{P}^1_k$ is the complex projective line and $L(d):=\mathcal{O}(d)$ with $d\in \mathbb{Z}$ it follows the global sections
$$H^0(C, L(d)) \cong k[x_0,x_1]_d$$
is the vector space of homogeneous polynomials $s$ in $x_0,x_1$ of degree $d$. If $d=2$ you get
$$s:=ax_0^2+bx_0x_1 +cx_1^2$$
and if you choose the open set $U_0:=D(x_0) \subseteq C$ you get
$$ s_{U_0}=(a+bt+ct^2)x_0^2:=f(t)x_0^2 =(t-u)(t-v)x_0^2$$
and to $s$ you get the divisor $[u]+[v]$ corresponding to the roots $u,v$ of the polynomial $f(t)$. More generally if $s_{U_0}:=f(t)x_0^d$ is a global section of $L(d)$ and you write
$$ s_{U_0}=\prod_i (t-u_i)^{l_i}x_0^d$$
you get the divisor
$$ \sum_i l_i[u_i]\in Cl(C)$$
in the group of Weil divisors on $C$. More generally if $\mathbb{P}^n$ is compex projective $n$-space it follows the global sections $H^0(\mathbb{P}^n, \mathcal{O}(d))$ is the vector space of homogeneous degree $d$ polynomials in $n+1$ variables. In Proposition II.6.4 in Hartshorne and the examples 6.5.1, 6.5.2 some examples are given.
The book is more than 700 pages long and give all details - again you should check these details if you need this construction. In Example II.8.20 in Hartshorne they write down the following exact sequence defining the tangent bundle of projective space $X:=\mathbb{P}^n_k$:
$$ 0 \rightarrow \mathcal{O}_X \rightarrow \mathcal{O}_X(1)^{n+1} \rightarrow T_X \rightarrow 0$$
and this sequence gives some information on the global sections of $T_X$:
$$ 0 \rightarrow H^0(X, \mathcal{O}_X) \rightarrow H^0(X, \mathcal{O}_X(1)^{n+1}) \rightarrow H^0(X, T_X) \rightarrow 0$$
and $H^0(X, \mathcal{O}_X(1)) \neq 0$. Hence the above sequence give non-trival global algebraic vector fields on projective space.
References: For algebraic varieties/schemes the relationship between Weil divisors/Cartier divisors, rational functions and invertible sheaves is explained in Hartshorne, Section II.6. For complex manifolds you may consult Griffiths/Harris "Principles of algebraic geometry", page 413 on the Gauss-Bonnet formulas II and III where the result (Theorem 5.3 above) is done for complex manifolds. They prove that the ith Chern class $c_i(E)$ of a holomorphic vector bundle $E$ on a complex manifold $M$ is Poincare dual to a "degeneracy cycle" $D_{k-r+1}$. They also prove the Gauss-Bonnet III formula that $c_n(M)=\chi(M)$. The Griffiths/Harris book is done in the language of compex manifolds and local coordinates and is easier to read for a person with background from differentiable/complex manifolds. The Chern classes and degeneracy cycles live in singular homology and cohomology groups of $M$. The Hartshorne approach is more abstract and algebraic.
Note: The books referred to above are in total $496 + 633 + 813 = 1942 $ pages and you can't yourself verify all results presented in the books. The results presented are classical results on characteristic classes and most of them are correct I believe.
